Vibration of double pendulum

I want to work out the natural frequencies (2) of this 2 DOF system through taking torques about the fixed point on the ceiling. I've done it using numerous fixed points and just cannot get the right answer.

2. Relevant equations
sum of torques = I . alpha

3. The attempt at a solution
Taking that cos theta = 1 and sin theta or tan theta = theta and acw moments are pos,

I want to work out the natural frequencies (2) of this 2 DOF system through taking torques about the fixed point on the ceiling. I've done it using numerous fixed points and just cannot get the right answer.View attachment 99206

2. Relevant equations
sum of torques = I . alpha

This equation is valid for a rigid body. Your system is not that, its parts move with relative to each other. The moment of inertia with respect the fixed point at the ceiling changes during the motion. Draw the FBD-s for both masses, and derive the accelerations in terms of the angles.